Theorem opncldf3 20700

Description: The values of the converse/inverse of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
opncldf.1	⊢ 𝑋 = ∪ 𝐽
opncldf.2	⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))

Assertion

Ref	Expression
opncldf3	⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))

Distinct variable groups:   𝑢,𝐽   𝑢,𝑋

Allowed substitution hints:   𝐵(𝑢)   𝐹(𝑢)

Proof of Theorem opncldf3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldrcl 20640	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		opncldf.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3		opncldf.2	. . . . . 6 ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))
4	2, 3	opncldf1 20698	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))
5	4	simprd 478	. . . 4 ⊢ (𝐽 ∈ Top → ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))
6	1, 5	syl 17	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))
7	6	fveq1d 6105	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵))
8	2	cldopn 20645	. . 3 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
9		difeq2 3684	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝐵))
10		eqid 2610	. . . 4 ⊢ (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))
11	9, 10	fvmptg 6189	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
12	8, 11	mpdan 699	. 2 ⊢ (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))‘𝐵) = (𝑋 ∖ 𝐵))
13	7, 12	eqtrd 2644	1 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ∪ cuni 4372   ↦ cmpt 4643  ^◡ccnv 5037  –1-1-onto→wf1o 5803  ‘cfv 5804  Topctop 20517  Clsdccld 20630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633

This theorem is referenced by: (None)